Functions are classified into two types under Calculus,
A linear function varies with a constant rate through its domain. Therefore, the overall rate of change of the function is same as the rate of change of a function at any point. However, the rate of change of function varies from point to point in case of non-linear functions. The nature of variation is based on the nature of the function. The rate of change of a function at a particular point is defined as a derivative of that particular function.
What is differentiation?
Differentiation can be defined as a derivative of a function regarding the independent variable and can be applied to measure the function per unit change in the independent variable.
Let y = f(x) be a function of x. Then, the rate of change of y per unit charge in x is given by
dy / dx
If the function f(x) undergoes an infinitesimal change of h near to any point x, the derivative of the function is defined as
When a function is denoted as y=f(x), the derivative is indicated by the following notations.
- D(y) or D[f(x)]is called Euler’s notation.
- dy/dx is called Leibniz’s notation.
- F’(x) is called as Lagrange’s notation.
Differentiation is the process of determining the derivative of a function at any point.
Some of the important Differentiation formulas in differentiation are as follows.
- If f(x) = tan (x), then f’(x) = sec2x
- If f(x) = cos (x), then f’(x) = -sin x
- If f(x) = sin (x), then f’(x) = cos x
- If f(x) = ln(x), then f’(x) = 1/x
- If f(x) = ex, then f’(x) = ex
- If f(x) = xn, where n is any fraction or integer, then f’(x) = nxn-1
- If f(x) = k, where k is a constant, then f’(x) = 0
Some of the basic differentiation rules that need to be followed are as follows.
Sum or Difference Rule
If the function is sum or difference of two functions, the derivative of the functions is the sum or difference of the individual functions, i.e.,
If f(x)=u(x)±v(x)f(x)=u(x)±v(x) then, f‘(x)=u‘(x)±v‘(x)f‘(x)=u‘(x)±v‘(x)
If the function f(x) is product of two functions u(x) and v(x), the derivative of the function is given by,
If the function f(x) is in the form of two functions [u(x)]/[v(x)], the derivative of the function is given by,
If a function y = f(x) = g(u) and if u = h(x), then,
This plays a major role in the method of substitution that helps to perform differentiation of composite functions.
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